146 XIII INTEGRATION
(To show that (1) implies (2) and (3), use Egoroff's theorem. To show that (3) im-
plies (1). show that (3) implies that there exists an increasing sequence of integrable
subsets A* of X and a subsequence (fnk) of the sequence (/„), such that ju,(Afc) tends to
p,(X) and f |/nfc| dp tends to 0, as k tends to -f oo.)
J Ak
10. Let X, Y be two locally compact spaces and TT : X -» Y a proper continuous mapping.
Let ft be a positive measure on X, and let v = TT(/LI) (Section 13.4, Problem 8).
(a) A mapping g of Y into a topological space is v-measurable if and only if g ° TT
is /^-measurable.
(b) A mapping g : Y -> R is v-integrable if and only if g ° TT is ju-integrable, and then
we have \ g dv = I (g o TT) dp. (Use Problem 12 of Section 13.8.)
(c) Show that the support of v is the closure of TT (
11. Let X be a locally compact space, u : X->X a continuous mapping, and /z a positive
measure on X. Suppose that for each ju-negligible set N, the inverse image u"1^)
is ^-negligible (cf. Problem 17). We shall write u~"(A) for (O-^A), for each integer
n ^ 1 and each subset A of X, and u° = lx -
00
If A is any jit-measurable subset of X, let ACnt = U «~"(A) be the set of points
which "enter A at least once," and let Arct = A n w~1(Aent) be the set of points of A
CO
which "return at least once to A". Also put Areunf = A n p| «~n(Aent), the set of
M = 0
points of A which "return to A infinitely often t" The set A is said to be wandering
(relative to u) if the sets u~n(A) (n 2> 0) are pairwise disjoint.
The mapping u is said to be incompressible if, for each ft-measurable subset A of X
such that A <= ^(A), the set u~l(A) — A is negligible; or equivalently if, for each
ft-measurable subset B of X such that w~J(B) c B, the set B - w^B) is negligible.
If u is not incompressible it is said to be compressible.
(a) Show that the following four properties are equivalent :
(a) u is incompressible.
(ft) The wandering sets (relative to u) are negligible.
(y) For each measurable set A, the set A — Aret is negligible.
(8) For each measurable set A, the set A — Aret mr is negligible.
(Observe that W^A^t) c Aent and that A — Aret is wandering.)
(b) Show that u is compressible if and only if there exists a measurable real-valued
function /on X such that/(w(x)) ^/(X) for all x eX and such that the set of points
x e X at which f(u(x)) >f(x) has measure >0. (To show that the condition is sufficient,
argue by contradiction to show that it implies that there exists a rational number r
for which the set of points x e X such that/(;c) < r <f(u(x)) is not negligible.) Deduce
that if un is incompressible for some integer n > 1, then u is incompressible.
(c) Suppose that X is compact and that the measure p, is invariant with respect to u
(cf. Section 13.4, Problem 8). Show that u is incompressible (" Poincare's recurrence
theorem ").
(d) Show that if u is incompressible, then there exists a negligible set N in X such
that, for every x $ N and every neighborhood V of x in X, there are infinitely many
integers n such that #"(#) 6 V. (Consider the sets U — Uretmr, as U runs through a
denumerable basis of open sets of X.)ost everywhere in X.